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I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way.

One type of model for $X_0(N)$ is just given by a single algebraic relation $\phi(j,j')$ between $j(\tau)$ and $j(N\tau)$. So given an elliptic curve $E$ of conductor $N$ I would really like to find $X, Y \in \mathbb Q(j)[j']/\phi$ such that $\mathbb Q(X,Y) \cong \mathbb Q(E)$.

Explicit parameterizations by other models of $X_0(N)$, for example using the weber f function instead of j, would also work for me. And in the light of the remark of David Loeffler below maybe even better. As long as there is a know expression for the $j$-invariant on that model.

I'm especially interested in the case where $N = 121$.

Maarten Derickx
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    Using the classical modular equation is likely to be a non-starter -- the coefficients of $\phi_{121}$ will be huge, far too big for any explicit computations! – David Loeffler Jan 23 '14 at 08:45
  • The answer at this question shows that at least there is an algorithm to find these maps: http://mathoverflow.net/questions/159126/is-the-modularity-theorem-currently-effective – Maarten Derickx Mar 03 '14 at 14:04

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There's a similar question on MO at:

Where can I find a comprehensive list of equations for small genus modular curves?

You might also check out

Y. Yang, "Defining equations of modular curves," Advances in Mathematics 204 (2006), 481-508.

Yang surveys the problem and then gives a method for finding defining equations with small coefficients by using explicit properties of some generalized Dedekind $\eta$ functions. He also gives lists of equations for $X_0(N)$ for $N \leq 50$. (So not $N=121$, but the techniques should work.) In his examples the coefficients are usually no more than $2$ digits.

Community
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Matt Papanikolas
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  • Yes I am aware that there exist several artikels on finding equations for modular curves. The problem is that I really also want equations for the modular parameterizations $X_0(N) \to E$. And I don't know of any paper that gives a systematic way of doing this. The paper by Y. Yang does this only for $X_0(37)$ by looking at quotients $X_0(37)/\iota$ where $\iota$ is some involution. But this technique will not generalize. – Maarten Derickx Feb 01 '14 at 12:35